Convex Lens: Real
Images

In this section we deal with a convex lens and its
ability to form real images. Real images are ones that can be
formed onto a screen. Examples of real images that we've all seen
include the images formed by slide projectors, movie projectors,
overhead projectors, cameras (the negative) and photographic
enlargers.

BUILD UP:

In the diagram below, we place a point light source at three
different places along the axis of our convex lens. The locations
are labeled A, B and C. Note that each of the
sources would be sending light rays out in all directions.

Now from position A, the light rays diverge with some
striking the lens and passing through it. On the other side, we
notice that the rays are still spreading out, they are still
diverging. But we notice that they aren't diverging as much as
before.

Perhaps at point B the light rays that strike the lens
emerge parallel to one another on the opposite side as illustrated
here.

We recognize this as exactly the condition described as
focal point, with B being
the focal point of our lens.

From position C, however, the light rays that strike the
lens come together on the other side of the lens, or
converge as shown here.

There is a unique place on the other side of the lens that the
light rays converge. If we were to put a screen of some sort
there, we would see a point of light. If the screen were now moved
away from the lens, the light rays would begin to diverge and the
light would spread out as if all the rays had come from that
point, not from point C. And if our eyes were somewhat
beyond the right-hand point to allow our lenses to focus
correctly, we would see a point source of light at that location.

What would happen if we defined other points, the same distance
from the lens as C, but above or below the axis? See the
setup below, followed by the result.

Notice how the light rays from the bottom converge to a point
above the axis while the rays from above the axis converge to a
point below the axis. Ctop represents the top of
an extended light source, the place where light rays that started
there come back together is the point labeled top', while
Cbotrepresents the bottom of an
extended light source. The place where these rays are converged is
represented by bot'. In the diagram which follows, we
overlay an arrow with its base at Cbot and its
tip at Ctop. Light from every point on this
object arrow would converge to a corresponding point in the line
between bot' and top'.

We have been able to reconstruct all of the rays which
originally started at the object and passed through the lens. This
reconstruction is the real image of the original object. A
screen placed at that location would have a pattern on it that
closely resembles the original object - the real image.

Not only were we able to reconstruct a pattern of light rays
that resembles the original pattern, but once they have been
brought together, the rays continue on out from there as if there
were a real object at that point.

But the pattern is upside down. With a little visualization,
you can imagine that the original arrow is rotated about the axis
and is now pointing into the paper. How will the image point? Yes,
out of the paper. The orientation of the image is both upside down
relative to the original object and reversed right-to-left. Such
an orientation is called inverted. The real image formed
by a convex les is inverted.

SIZE:

If you observe closely, you will notice that the length of the
object arrow in our diagram is slightly greater than the length of
the image arrow. There must be some sort of size relationship
which we must account for in any theory we develop.

PRINCIPLE RAYS:

In this drawing we see any number of light rays, the red ones
coming from the head of the object arrow and the blue ones coming
from the tail. Even with only five rays coming from each point you
can imagine how difficult it's going to be to calculate what
direction to take each light ray after it strikes the lens. And to
think that light does this automatically!

There are two light rays that we have a good idea about based
on our definition of focal point. Even though all the rays will
behave in a similar manner, we can use them to show us the general
pattern.

In the diagrams below, we concentrate our attention on the
light coming from the tip of the object arrow. Note that we've now
marked the locations of the focal points, one on each side of the
lens, with the letter f.

Of all the light rays, the one which started out parallel to
the axis will pass through the lens and be bent towards the focal
point on the other side. It doesn't stop there but goes on until
it runs into something new.

A light ray which goes through the first focal point will be
bent until it emerges parallel to the axis. This is the
complementary behavior to the previous one. Note that we know this
behavior, while we can only approximate the behavior of other
light rays.

With both of these rays drawn, we see that they intersect at a
particular place on the other side of the lens. Now if we had many
more rays in our diagram, where would they intersect?

They all intersect at the same point. We don't need all of them
to locate that point, but once we've located it, we can be assured
that the rest will follow. So we can use just the two rays to
locate the point and we call these the principle rays - the
one that was originally parallel to the axis and the one which
passed through the focal point.

The light from the center of the object will converge at the
same distance on the other side of the lens, but will be closer to
the axis. The light from the bottom of the object will likewise be
the same distance, but will focus on the axis. If we join these
points and draw in a complete image, we get the diagram below:

A screen placed at this location will show an entire image that
looks remarkably like the original object, but it is inverted. But
we only need the three principle rays to make the same conclusion
about size and location shown here:

Things look slightly different, but better, when we see white
light against a black background:

ANOTHER PRINCIPLE RAY:

When we consider a lens, we note that the sides of the lens
near the center are approximately parallel. Thus light would
behave very much like going through a piece of glass. In this
diagram we remind ourselves about the path that light would take
through a fairly thick piece of glass:

The light ray that emerges on the other side of the glass is
parallel to the one which entered. It is displaced somewhat, but
is going in the same general direction.

If the glass is thin, the displacement is minor. If it is thin
enough, the displacement can essentially be ignored and we can
approximate the path as simply a straight line. This approximation
is often referred to as the thin lens approximation.

If our lens is thin - its thickness is small compared to the
diameter, or thin compared to the distance between the object and
the lens - we can see a ray going virtually straight through the
center and then passing through the corresponding point on the
image. This is shown here as a third principle ray.

OR

We mostly use the first two rays, the one parallel to the axis
and the one that passes through the focal point, to determine the
image location. This third ray will also arrive at the same point,
but is dependent on an approximation to be as accurate as the
others.

SUMMARY TO DATE:

An object placed fairly far from a convex lens will form a
real image on the other side of the lens.

The real image will be inverted - upside down and backwards
right to left.

The real image can be formed onto a screen like in a slide
projector.

The real image may be a different size than the
object.

SIGNS:

For lenses, we develop a sense of signs in our mathematics. It
comes from the direction that light travels through the lens. In
the diagram which follows, note that the positive sense of things
occurs when light starts on one side and converges on the other.